a) Since
\(\sup _{n\in \mathbb {N}}\int _{G_{m}}\left|K_{n}\left( x\right)\right|d\mu \left( x\right) \le c<\infty , \) 
we obtain that
\(\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left|\sigma _{n_{k}}a\left( x\right) \right|}{M_{\left|n_{k}\right|}^{1/p-2}}\) 
is bounded from \(L_{\infty }\)  to \(L_{\infty }.\)  According to Lemma REF  we find that the proof of
Theorem REF  will be complete, if we show that
\(\int _{\overline{I_{N}}}\left|\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\sigma _{n_{k}}a\left( x\right) }{M_{\left|n_{k}\right|}^{1/p-2}}\right|^{p}<c<\infty ,\) 
for every \(p\) -atom \(a,\)  with support\(\ I\)  and \(\mu \left( I\right)=M_{N}^{-1}.\)  We may assume that \(I=I_{N}.\)  It is easy to see that \(\sigma _{n_{k}}\left( a\right) =0\)  when \(n_{k}\le M_{N}.\)  Therefore, we can
suppose that \(n_{k}>M_{N}\) .
Since \(\left\Vert a\right\Vert _{\infty }\le M_{N}^{1/p}\)  we find that
\( &&\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left|\sigma _{n_{k}}a\left( x\right) \right|}{M_{\left|n_{k}\right|}^{1/p-2}}\le \frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}}{M_{\left|n_{k}\right|}^{1/p-2}}\int _{I_{N}}\left|a\left(t\right) \right|\left|K_{n_{k}}\left( x-t\right) \right|d\mu \left( t\right)  \\&\le &\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left\Vert a\right\Vert _{\infty }}{M_{\left|n_{k}\right|}^{1/p-2}}\int _{I_{N}}\left|K_{n_{k}}\left( x-t\right) \right|d\mu \left(t\right) \\&\le &\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}M_{N}^{1/p}}{M_{\left|n_{k}\right|}^{1/p-2}}\int _{I_{N}}\left|K_{n_{k}}\left( x-t\right) \right|d\mu \left( t\right) \\&\le &M_{\left\langle n_{k}\right\rangle }^{1/p-2}M_{\left|n_{k}\right|}^{2}\int _{I_{N}}\left|K_{n_{k}}\left( x-t\right)\right|d\mu \left( t\right) . \) 
Without loss the generality we may assume that \(i<j\) . Let \(x\in I_{N}^{i,j}\ \) and \(j<\left\langle n_{k}\right\rangle .\)  Then \(x-t\in I_{N}^{i,j}\)  for \(t\in I_{N}\)  and, according to Lemma REF , we obtain that
\(\left|K_{M_{l}}\left( x-t\right) \right|=0,\text{ \ for all }\left\langle n_{k}\right\rangle \le \text{ }l\le \left|n_{k}\right|.\) 
By applying (REF ) and (REF ) in Lemma REF , for \( x \in I_{N}^{i,j},\text{ \ }0\le i<j<\left\langle n_{k}\right\rangle \)  we get that
\( &&\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left|\sigma _{n_{k}}a\left( x\right) \right|}{M_{\left|n_{k}\right|}^{1/p-2}}\le M_{\left\langle n_{k}\right\rangle }^{1/p-2}M_{\left|n_{k}\right|}^{2}\overset{\left|n_{k}\right|}{\underset{l=\left\langle n_{k}\right\rangle }{\sum }}\int _{I_{N}}\left|K_{M_{l}}\left( x-t\right) \right|d\mu \left( t\right) =0.\text{ }\) 
Let \(x\in I_{N}^{i,j},\,\) where \(\left\langle n_{k}\right\rangle \le j\le N.\)  Then, in the view of Lemma REF , we have that
\(\int _{I_{N}}\left|K_{n_{k}}\left( x-t\right) \right|d\mu \left(t\right) \le \frac{cM_{i}M_{j}}{M_{N}^{2}}.\) 
By using again (REF ) we find that
\( \frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left|\sigma _{n_{k}}a\left( x\right) \right|}{M_{\left|n_{k}\right|}^{1/p-2}} &\le &\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}M_{N}^{1/p}}{M_{\left|n_{k}\right|}^{1/p-2}}\int _{I_{N}}\left|K_{n_{k}}\left( x-t\right) \right|d\mu \left(t\right) \\&\le &\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}M_{N}^{1/p}}{M_{\left|n_{k}\right|}^{1/p-2}}\frac{M_{i}M_{j}}{M_{N}^{2}}\le M_{\left\langle n_{k}\right\rangle }^{1/p-2}M_{i}M_{j}. \) 
By combining (REF ) and (REF )-(REF ) we get that
\(&&\int _{\overline{I_{N}}}\left|\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left|\sigma _{n_{k}}a\left( x\right) \right|}{M_{\left|n_{k}\right|}^{1/p-2}}\right|^{p}d\mu \\&=&\overset{N-2}{\underset{i=0}{\sum }}\overset{N-1}{\underset{j=i+1}{\sum }}\int _{I_{N}^{i,j}}\left|\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left|\sigma _{n_{k}}a\left( x\right) \right|}{M_{\left|n_{k}\right|}^{1/p-2}}\right|^{p}d\mu \\ &+&\overset{N-1}{\underset{i=0}{\sum }}\int _{I_{N}^{k,N}}\left|\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left|\sigma _{n_{k}}a\left( x\right)\right|}{M_{\left|n_{k}\right|}^{1/p-2}}\right|^{p}d\mu \\&\le &\overset{\left\langle n_{k}\right\rangle -1}{\underset{i=0}{\sum }}\overset{N-1}{\underset{j=\left\langle n_{k}\right\rangle }{\sum }}\int _{I_{N}^{i,j}}\left|\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left|\sigma _{n_{k}}a\left( x\right) \right|}{M_{\left|n_{k}\right|}^{1/p-2}}\right|^{p}d\mu \\&&+\overset{N-2}{\underset{i=\left\langle n_{k}\right\rangle }{\sum }}\overset{N-1}{\underset{j=i+1}{\sum }}\int _{I_{N}^{i,j}}\left|\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left|\sigma _{n_{k}}a\left( x\right) \right|}{M_{\left|n_{k}\right|}^{1/p-2}}\right|^{p}d\mu \\&&+\overset{N-1}{\underset{i=0}{\sum }}\int _{I_{N}^{i,N}}\left|\frac{M_{\left\langle n_{k}\right\rangle }^{1/p-2}\left|\sigma _{n_{k}}a\left( x\right) \right|}{M_{\left|n_{k}\right|}^{1/p-2}}\right|^{p}d\mu \) 
\(&\le &\overset{\left\langle n_{k}\right\rangle -1}{\underset{i=0}{\sum }}\overset{N-1}{\underset{j=\left\langle n_{k}\right\rangle }{\sum }}\int _{I_{N}^{i,j}}\left|M_{\left\langle n_{k}\right\rangle }^{1/p-2}M_{i}M_{j}\right|^{p}d\mu +\overset{N-2}{\underset{i=\left\langle n_{k}\right\rangle }{\sum }}\overset{N-1}{\underset{j=i+1}{\sum }}\int _{I_{N}^{i,j}}\left|M_{\left\langle n_{k}\right\rangle }^{1/p-2}M_{i}M_{j}\right|^{p}d\mu \\&&+\overset{N-1}{\underset{i=0}{\sum }}\int _{I_{N}^{i,N}}\left|M_{\left\langle n_{k}\right\rangle }^{1/p-2}M_{i}M_{N}\right|^{p}d\mu \\&\le &c_{p}M_{\left\langle n_{k}\right\rangle }^{1-2p}\overset{\left\langle n_{k}\right\rangle -1}{\underset{i=0}{\sum }}\overset{N-1}{\underset{j=\left\langle n_{k}\right\rangle }{\sum }}\frac{\left( M_{i}M_{j}\right)^{p}}{M_{j}}+c_{p}M_{\left\langle n_{k}\right\rangle }^{1-2p}\overset{N-2}{\underset{i=\left\langle n_{k}\right\rangle }{\sum }}\overset{N-1}{\underset{j=i+1}{\sum }}\frac{\left( M_{i}M_{j}\right) ^{p}}{M_{j}} \\&&+c_{p}M_{\left\langle n_{k}\right\rangle }^{1-2p}\underset{i=0}{\sum }\frac{\left( M_{i}M_{N}\right) ^{p}}{M_{N}} \\&\le &c_{p}M_{\left\langle n_{k}\right\rangle }^{1-2p}\overset{\left\langle n_{k}\right\rangle }{\underset{i=0}{\sum }}M_{i}^{p}\overset{N-1}{\underset{j=\left\langle n_{k}\right\rangle +1}{\sum }}\frac{1}{M_{j}^{1-p}}+M_{\left\langle n_{k}\right\rangle }^{1-2p}\overset{N-2}{\underset{i=\left\langle n_{k}\right\rangle }{\sum }}M_{i}^{p}\overset{N-1}{\underset{j=i+1}{\sum }}\frac{1}{M_{j}^{1-p}} \\&&+c_{p}\overset{N-1}{\underset{i=0}{\sum }}\frac{M_{i}^{p}}{M_{N}^{p}} \\&\le &c_{p}M_{\left\langle n_{k}\right\rangle }^{1-2p}M_{\left\langle n_{k}\right\rangle }^{p}\frac{1}{M_{\left\langle n_{k}\right\rangle }^{1-p}}+c_{p}M_{\left\langle n_{k}\right\rangle }^{1-2p}\overset{N-2}{\underset{i=\left\langle n_{k}\right\rangle }{\sum }}\frac{1}{M_{i}^{1-2p}}+c_{p}\le c_{p}<\infty .\) 
The proof of the a) part is complete.
b) Let
\(\left\lbrace  n_{k}:k\ge 0\right\rbrace  \)  be a sequence of positive numbers,
satisfying condition (REF ). Then
\(\sup _{k\in \mathbb {N}}\frac{M_{\left|n_{k}\right|}}{M_{\left\langle n_{k}\right\rangle }}=\infty . \) 
Under condition (REF ) there exists a sequence \(\left\lbrace  \alpha _{k}:\text{ }k\ge 0\right\rbrace  \subset \left\lbrace  n_{k}:\text{ }k\ge 0\right\rbrace  \)  such
that \(\alpha _{0}\ge 3\)  and
\(\sum _{k=0}^{\infty }\frac{M_{\left\langle \alpha _{k}\right\rangle }^{\left(1-2p\right) /2}\Phi ^{p/2}\left(\alpha _{k}\right)}{M_{\left|\alpha _{k}\right|}^{\left( 1-2p\right)/2}}<c<\infty . \) 
Let       
\(f^{\left( n\right) }=\sum _{\left\lbrace  k;\text{ }\left|\alpha _{k}\right|<n\right\rbrace  }\lambda _{k}a_{k},\) 
where
\(\lambda _{k}=\frac{\lambda M_{\left\langle \alpha _{k}\right\rangle }^{\left( 1/p-2\right) /2}\Phi ^{1/2}\left(\alpha _{k}\right)}{M_{\left|\alpha _{k}\right|}^{\left(1/p-2\right) /2}}\) 
and
\(a_{k}=\frac{M_{\left|\alpha _{k}\right|}^{1/p-1}}{\lambda }\left(D_{M_{\left|\alpha _{k}\right|+1}}-D_{M_{\left|\alpha _{k}\right|}}\right) .\) 
B applying Lemma REF  we can conclude that \(f\in H_{p}.\)
It is evident that
\(\widehat{f}(j)=\left\lbrace \begin{array}{l}M_{\left|\alpha _{k}\right|}^{1/2p}M_{\left\langle \alpha _{k}\right\rangle }^{\left( 1/p-2\right) /2}\Phi ^{1/2}\left(\alpha _{k}\right),\,\,\text{ } \\\text{if \,\,}j\in \left\lbrace  M_{\left|\alpha _{k}\right|},...,\text{ ~}M_{\left|\alpha _{k}\right|+1}-1\right\rbrace  ,\text{ }k=0,1,2..., \\0\text{ },\text{ \,\qquad \,\ \ \ \ \,\ \ \ \ \ }\\\text{\ if \,\,\,}j\notin \bigcup \limits _{k=0}^{\infty }\left\lbrace  M_{\left|\alpha _{k}\right|},...,\text{ ~}M_{\left|\alpha _{k}\right|+1}-1\right\rbrace  .\text{ }\end{array}\right. \) 
Moreover,
\(\frac{\sigma _{_{\alpha _{k}}}f}{\Phi \left(\alpha _{k}\right)}=\frac{1}{\alpha _{k}\Phi \left(\alpha _{k}\right)}\sum _{j=1}^{M_{\left|\alpha _{k}\right|}}S_{j}f+\frac{1}{\alpha _{k}\Phi \left(\alpha _{k}\right)}\sum _{j=M_{\left|\alpha _{k}\right|}+1}^{\alpha _{k}}S_{j}f:=I+II.\) 
Let \(M_{\left|\alpha _{k}\right|}<j\le \alpha _{k}.\)  Then, by
applying (REF ) we get that
\(S_{j}f=S_{M_{\left|\alpha _{k}\right|}}f+M_{\left|\alpha _{k}\right|}^{1/2p}M_{\left\langle \alpha _{k}\right\rangle }^{\left(1/p-2\right)/2}\Phi ^{1/2}\left(\alpha _{k}\right)\left( D_{j}-D_{M_{\left|\alpha _{k}\right|}}\right). \) 
By using (REF ) we can rewrite \(II\)  as
\(II &=&\frac{\alpha _{k}-M_{\left|\alpha _{k}\right|}}{\alpha _{k}\Phi \left(\alpha _{k}\right)}S_{M_{\left|\alpha _{k}\right|}}f+\frac{M_{\left|\alpha _{k}\right|}^{1/2p}M_{\left\langle \alpha _{k}\right\rangle }^{\left(1/p-2\right)/2}}{\alpha _{k}\Phi ^{1/2}\left(\alpha _{k}\right)}\sum _{j=M_{\left|\alpha _{k}\right|}}^{\alpha _{k}}\left( D_j-D_{M_{\left|\alpha _{k}\right|}}\right) \\&:=&II_{1}+II_{2}.\) 
Since (for details see e.g. [1]} and [2]})
\(\left\Vert S_{M_{\left|\alpha _{k}\right|}}f\right\Vert _{weak-L_{p}}\le c_{p}\left\Vert f\right\Vert _{H_{p}}\) 
we obtain that
\(&&\left\Vert II_{1}\right\Vert _{weak-L_{p}}^{p}\le \left(\frac{\alpha _{k}-M_{\left|\alpha _{k}\right|}}{\alpha _{k}\Phi \left(\alpha _{k}\right)}\right)^{p}\left\Vert S_{M_{\left|\alpha _{k}\right|}}f\right\Vert _{weak-L_{p}}^{p} \\&\le & \left\Vert S_{M_{\left|\alpha _{k}\right|}}f\right\Vert _{weak-L_{p}}^{p}\le c_{p}\left\Vert f\right\Vert _{H_{p}}^{p}<\infty .\) 
By using part a) of Theorem REF  we find that
\(\left\Vert I\right\Vert _{weak-L_{p}}^{p}=\left( \frac{M_{\left|\alpha _{k}\right|}}{\alpha _{k}\Phi \left(\alpha _{k}\right)}\right) ^{p}\left\Vert \sigma _{M_{\left|\alpha _{k}\right|}}f\right\Vert _{weak-L_{p}}^{p}\le c_{p}\left\Vert f\right\Vert _{H_{p}}^{p}<\infty .\) 
Let \(x\in \)  \(I_{_{\left\langle \alpha _{k}\right\rangle +1}}^{\left\langle \alpha _{k}\right\rangle -1,\left\langle \alpha _{k}\right\rangle }.\)  Under
condition (REF ) we can conclude that \(\left\langle \alpha _{k}\right\rangle \ne \left|\alpha _{k}\right|\)  and \(\left\langle \alpha _{k}-M_{\left|\alpha _{k}\right|}\right\rangle =\left\langle \alpha _{k}\right\rangle .\)  Since
\(D_{j+M_{n}}=D_{M_{n}}+\psi _{M_{n}}D_{j}=D_{M_{n}}+r_{n}D_{j},\text{ when }\,\,j<M_{n} \) 
if we apply estimate (REF ) in Lemma REF  for \(II_{2}\)  we obtain
that
\(\left|II_{2}\right|&=&\frac{M_{\left|\alpha _{k}\right|}^{1/2p}M_{\left\langle \alpha _{k}\right\rangle }^{\left( 1/p-2\right) /2}}{\alpha _{k}\Phi ^{1/2}\left(\alpha _{k}\right)}\left|\sum _{j=1}^{\alpha _{k}-M_{\left|\alpha _{k}\right|}}\left( D_{j+M_{\left|\alpha _{k}\right|}}-D_{M_{\left|\alpha _{k}\right|}}\right) \right|\\&=&\frac{M_{\left|\alpha _{k}\right|}^{1/2p}M_{\left\langle \alpha _{k}\right\rangle }^{\left( 1/p-2\right) /2}}{\alpha _{k}\Phi ^{1/2}\left(\alpha _{k}\right)}\left|\psi _{M_{\left|\alpha _{k}\right|}}\sum _{j=1}^{\alpha _{k}-M_{\left|\alpha _{k}\right|}}D_{j}\right|\\&\ge &\frac{c_{p}M_{\left|\alpha _{k}\right|}^{1/2p-1}M_{\left\langle \alpha _{k}\right\rangle }^{\left( 1/p-2\right) /2}}{\Phi ^{1/2}\left(\alpha _{k}\right)}\left( \alpha _{k}-M_{\left|\alpha _{k}\right|}\right) \left|K_{\alpha _{k}-M_{\left|\alpha _{k}\right|}}\right|\\&\ge &\frac{c_{p}M_{\left|\alpha _{k}\right|}^{1/2p-1}M_{\left\langle \alpha _{k}\right\rangle }^{\left( 1/p+2\right) /2}}{\Phi ^{1/2}\left(\alpha _{k}\right)}.\) 
It follows that
\(&&\left\Vert II_{2}\right\Vert _{weak-L_{p}}^{p} \\&\ge &c_{p}\left( \frac{M_{\left|\alpha _{k}\right|}^{\left(1/p-2\right) /2}M_{\left\langle \alpha _{k}\right\rangle }^{\left(1/p+2\right) /2}}{\Phi ^{1/2}\left(\alpha _{k}\right)}\right) ^{p}\mu \left\lbrace  x\in G_{m}:\left|IV_{2}\right|\ge c_{p}M_{\left|\alpha _{k}\right|}^{\left(1/p-2\right) /2}M_{\left\langle \alpha _{k}\right\rangle }^{\left(1/p+2\right) /2}\right\rbrace  \\&\ge &c_{p}\frac{M_{\left|\alpha _{k}\right|}^{1/2-p}M_{\left\langle \alpha _{k}\right\rangle }^{1/2+p}\mu \left\lbrace I_{_{\left\langle \alpha _{k}\right\rangle +1}}^{\left\langle \alpha _{k}\right\rangle -1,\left\langle \alpha _{k}\right\rangle }\right\rbrace }{\Phi ^{p/2}\left(\alpha _{k}\right)}\ge \frac{c_{p}M_{\left|\alpha _{k}\right|}^{1/2-p}}{M_{\left\langle \alpha _{k}\right\rangle }^{1/2-p}\Phi ^{p/2}\left(\alpha _{k}\right)}.\) 
Hence, for large \(k\) ,
\(&&\left\Vert \sigma _{\alpha _{k}}f\right\Vert _{weak-L_{p}}^{p} \\&\ge &\left\Vert II_{2}\right\Vert _{weak-L_{p}}^{p}-\left\Vert II_{1}\right\Vert _{weak-L_{p}}^{p}-\left\Vert I\right\Vert _{weak-L_{p}}^{p}\\&\ge &\frac{1}{2}\left\Vert II_{2}\right\Vert _{weak-L_{p}}^{p}\ge \frac{c_{p}M_{\left|\alpha _{k}\right|}^{1/2-p}}{2M_{\left\langle \alpha _{k}\right\rangle }^{1/2-p}\Phi ^{p/2}\left(\alpha _{k}\right)}\rightarrow \infty ,\text{ as }k\rightarrow \infty .\) 
The proof is complete.
